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SL 5.3—Differentiating polynomials, n E Z - IB Questionbank | RevisionDojo

Practice SL 5.3—Differentiating polynomials, n E Z with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ ,.

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

SL & HLPaper 1

Consider the function $f(x) = \sin(x) + \cos(x)$ , where $x$ is in radians.

Find the derivative of the function $f(x) = \sin(x) + \cos(x)$ .

[2]

Determine the critical points of the function $f(x) = \sin(x) + \cos(x)$ in the interval $[0, 2\pi]$ .

[3]

Classify the critical points found in part (b) as local maxima, minima, or inflection points.

[3]

Question 3

SLPaper 1

A civil engineer is designing a sloped drainage canal, modeled by the function $f(x)=bx^3−8x^2$ , where $x$ represents the horizontal distance along the canal (in meters) from its starting point, and $f(x)$ gives the depth of the canal at any point.

The slope of the canal at various locations impacts the speed of water flow, so it's essential to ensure an appropriate gradient. At a specific point $P(2,y)$ along the canal, the slope of the canal is measured to be $4$ . This information will help the engineer determine the value of the constant $b$ for the correct canal design.

Find the expression for the slope of the canal at any point $x$ by calculating $f'(x)$ .

[2]

Given that the slope of the canal at point $P(2,y)$ is $4$ , find the value of the constant b needed to achieve this slope at point P.

[2]

Question 4

SLPaper 1

Consider the function $f(x) = 3x^4 - 8x^3 + 6x^2 - 4x + 1$ .

Find the first derivative $f'(x)$ .

[2]

Find the second derivative $f''(x)$ .

[2]

Find the points of inflection of the function.

[3]

Question 5

SL & HLPaper 1

Consider the function $h(x) = x^3 - 6x^2 + 9x + 1$ .

Determine the critical points and classify them as local maxima, minima, or points of inflexion.

[6]

Question 6

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 7

SL & HLPaper 1

Given the function $g(x) = \tan(x)$ , where $x$ is in radians and $x \neq \frac{\pi}{2} + n\pi$ , $n \in \mathbb{Z}$ .

Find the derivative of the function $g(x) = \tan(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = \tan(x)$ at the point where $x = \frac{\pi}{4}$ .

[4]

Question 8

HLPaper 1

This question involves the introduction of differential calculus and focuses on finding the derivative of a function using first principles and applying basic differentiation rules.

Consider the function $f(x) = 3x^2 - 2x + 4$ . Using the definition of the derivative, find $f'(x)$ .

[5]

Using the result from part 1, find the slope of the tangent to the curve $y = f(x)$ at the point where $x = 1$ .

[2]

Find the equation of the tangent line to the curve $y = f(x)$ at the point where $x = 1$ .

[3]

Determine the x-coordinates where the tangent to the curve $y = f(x)$ is horizontal.

[2]

Question 9

HLPaper 1

Consider the function $f(x) = x^2 \sin(x) + \cos(x)$ .

Find the derivative of the function $f(x) = x^2 \sin(x) + \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Find the equation of the normal line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Question 10

SLPaper 1

An architect is designing a roller coaster with a section of track modeled by the function $f(x)=6x^3−5x^2+2x−7$ , where $x$ represents the horizontal distance along the track (in meters), and $f(x)$ gives the height of the track at any point.

To ensure safety and thrill, the architect needs to know the slope of the track at specific points. The slope at $x=2$ will help the architect understand the steepness at this part of the ride.

Find the expression for the slope of the roller coaster track at any point $x$ .

[3]

Calculate the the steepness of the ride at $x=2$ .

[1]

SL 5.3—Differentiating polynomials, n E Z Questionbank